Buckling Analysis of Concrete Spherical Shells

8/11/2019 Buckling Analysis of Concrete Spherical Shells

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1Introduction

In some simple two-dimensional structures it issufficient to determine only the lowest value of the load atwhich buckling commences. However, in the case of shells,it may be also necessary to investigate the postbucklingbehavior because it has an important bearing on themagnitude of thefailure load.

The importance of the postbuckling behavior, whichthe small deflection theory of buckling is not capable ofpredicting, was discovered as a consequence of attempts tocorrelateexperimentalresultswithanalyticalpredictions.

Poor correlation between the results of theory andexperiment existswhen both principalmembraneforcesarecompressive, as in the case of cylindrical shells under axialcompressive load; cylindrical shells under distributed loadnormal to the surface, which causes bending; and domesunder inward radial pressure. If both the principalmembrane forces are compressive, they tend to increase

with deformation of the shell.After the initial buckling, theshell can only transmit loads smaller than the initialbuckling load. This is particularly true for concrete shellsbecause of creep and deviation of the actual shape of theshellfrom theassumedtheoretical surface.

Good correlation exists when one of the principalmembraneforces is tensile. Ifoneof these forcesis tensile, ittends to return the shell back to its original position, thusenabling it to carry loads greater than the initial bucklingload.

In general, the value of the buckling load depends onshell geometry, type of restraint at boundary, materialproperties of shell, the location of reinforcing steel, and the

type of load.

633

I. Mekjavi

ISSN 1330-3651

UDC/UDK 624.044:624.012.4.074.43

BUCKLING ANALYSIS OF CONCRETE SPHERICAL SHELLS

Ivana Mekjavi

This paperpresents a bucklinganalysisfor several notable concrete thin shellsaround theworld.Anapproachwhichtakes into account thelarge deflection andplasticity effects was performed using Sofistik software to estimate the buckling load.Ageometrically non-linearanalysis of these structures with and withoutgeometrical imperfections was performed. To take into account the possible plastification of the material a materially non-linear analysis was performedsimultaneously with the geometrically non-linear analysis. The buckling analysis of concrete spherical shells shows that including only one kind of non-linearitydoes notgive a realisticsituationand onlytheircombination results in thedecreasing ofultimatefailure load.

Keywords: buckling analysis, concrete shells, spherical shells, geometrically non-linear analysis, materially non-linear analysis

Subject review

U ovom radu je prikazana analiza sfernih ljusaka u svijetu. Pristup kojim se uzimajuijenjen je uporabom programa Sofistik radi procjene opte . Provela se geometrijski nelinearna analiza ovih

konstrukcija sa i bez geometrijske nost plastifikacije se provela materijalno nelinearna analizaistodobno s geometrijski nelinearnom analizom. se

o stanje te da samo njihovo kombiniranje .

izboavanja nekoliko znaajnih betonskih u obzir velike deformacije iuinci plastinosti prim reenja izboavanja

imperfekcije. Da bi se uzela u obzir mogu materijala takoerAnaliza izboavanja betonskih sfernih ljusaka pokazala je da ukljuujui samo jedan tip nelinearnosti ne

dobiva realn dovodi do smanjenja graninog optereenja sloma

Kljune rijei: analiza izb betonske ljuske, sferne ljuske, geometrijski nelinearna analiza, materijalno nelinearna analizaoavanja,

Pregledni lanak

Analiza izboavanja betonskih sfernih ljusaka

Tehni ki vjesnik 18, 4(2011) 633-639,

Analiza izboavanja betonskih sfernih ljusaka

2Bucklingofdomes

The classical analysis for the bifurcation buckling ofspherical domes under axisymmetrical radial pressure

waslongagofoundtobe[1,2]

qcr

)1(3

2

22

2

r

Ehqcr (1)

where: modulus of elasticity, MPa thicknessof the shell, m radiusof the sphere,m Poisson's .

Thecorrespondingcritical stress is therefore

Ehr ratio

.)1(32 2

r

Eh

h

rqcrcr ( )2

When theuniformexternalpressure exceeds , given

by equation (1), i.e., > , the system is unstable. If =

the spherical shell is in neutral equilibrium for smalldisplacements. If < theshellis instableequilibrium.

The value of critical stress defined by equation (2),which is based upon the small-displacement theory, oftendoes not agree with experimental data. However, the realvalue of the critical stress is much lower than this.Experimental results have repeatedly shown criticalstresses of as low as 10 percent of the classical, i.e., of thatpredicted by the small deflection theory [3]. By the use oflarge deflection theory of buckling and plasticity effects,results may be obtained which closely approximateexperimental values. A spherical shell under radial external

q q

q q q q

q q

cr

cr cr

cr


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634 Technical Gazette 18, 4(2011) 633-639,

I. Mekjavi

pressure will buckle suddenly or oilcan by leaping to alower state of energy at a stress far below that given by theclassic value.

For practical applications we can use the followingempiricalformulafor calculating [1]:

" "

qcr

These structures were built before the use of computers.Prior to the availability of computers and the finite elementmethod, shells were analyzed using approximate methodswhich forcedthedesigner todevelop an intuitivefeel forthestructural behaviour of the shell, which is sometimesmissing with theuncriticaluseof computers.

Structural analysis and the optimization study of theseshellswerealso performedusingSofistiksoftwarein[5].

The Kresge Auditorium at MIT, designed by a notedmodernist architect, Eero Saarinen, consists of a one-eighthspherical segment dome-shaped concrete roof enclosing atriangular area approximately 49 m (160 ft) on a side. Thedome is entirely supported on three points at the vertices ofthe triangle. The total weight of the roof is approximately1500 tons, and the thickness of the roof shell is 8,9 cm (3,5in) which is increased near the edge beams up to 14 cm. The8,9 cm(3,5in)concreteshell is covered with5 1cm (2in) of

glass fiberboard and a second non-structural layer oflightweight concrete 5,1cm(2 in) thick.Additions hadto bemade to this structure, since Saarinen's sculptural cutting ofthe shell created severe edge disturbances to the membranestresses in the shell that had to be counteracted by an edgebeam (45,7 cm (18 in) height). There were also largestresses created at the three points of support. These werereinforced with tapered H-shaped steel ribs, which in turnwere connected to a steel hinge allowing for movement. Inthe end, after the formwork was removed it was discoveredthat the edges were deflecting an unacceptable amount(clearly well over 12,7 cm (5 in)) due to uncontrolled creep.Additional supports were added in the form of (4-by-9-in)steel tubes spaced at 3,35 m (11 ft), which are also used tosupport thewindow wall [6].

The problems with this building did not end with thesolution of the structural problems. The shell was difficultand unusual to construct, and significant difficulties wereencountered in concrete placement (poor consolidation),protection of the reinforcing steel (inadequate concretecover) and above all in the waterproofing the roof of thebuilding. The satisfactory solution of these problems had towait until decades after the commissioning of the buildingandthroughseveral trials ofdifferentroofingprocedures.

The original neoprene roofing was later replaced withlead-coated copper roofing and then copper roofing. Therepair of the construction was costly and forced the closure

of thebuildingfor a fewmonths.

The Ehime Public Hall in Matsuyama, Japan, designedby Japanese engineers, Tange and Tsuboi, is a shallowspherical inclined shell supported by 20 columns.A ring isprovided around the base between columns. The thicknessoftheshell is8 cmwith a diameterof 49,35 m and a riseof 7matthecrown[7].

The Het Evoluon in Eindhoven was the last majorproject of the Netherlands designer Louis Kalff. Thebuilding is unique due to its resemblance to a landed flying

3.1KresgeAuditorium

3.2EhimePublicHall

3.3

HetEvoluon

,

Buckling analysis of concrete spherical shells

.

Using the Sofistik finite element program that solves

large-scale structural analysis problems, several sphericalshellstructureswere examined.Figs 1-3 show some of the remarkable early shells for

the Kresge MITauditorium in Boston, Ehime Public HallinMatsuyamaandHetEvoluoninEindhoven.

2

cr )3,0(

400

/07,01

20

20175,01

r

hE

hrq

(3)

whichgivessatisfactory results for, where is the angle between the axis of

rotationandthe dome edge.Here an approach which takes into account the large

deflection and plasticity effects was performed by usingSofistik software [4]. This approach is based onminimization of the energy of the system, i.e., energymethods. It also permits computation of the lower as well astheupperboundaryof thebuckling load, andis applicable toalltypesofshells.

and

3Analyzedconcretesphericalshells

2000/400 hr 6020

Figure 1Kresge-MITAuditorium, Boston, USA(Saariner, 1954)

Figure 2Ehime Public Hall, Matsuyama, Japan (Tange&Tsuboi)

Figure 3Het Evoluon, Eindhoven, Netherlands (Kalff, 1966)


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636 Technical Gazette 18, 633-639

eigenvalue determination gives a buckling factorof 1,08 forthe first buckling mode on deformed structure withgeometrical imperfection, as shown in Fig. 7. Forcomparison, the buckling factor of 99,12 is obtained for thefirst buckling mode on undeformed structure with the snowload as primary load case i.e. under the stresses of theprimary load case (Fig. 5). The geometric stiffness from theprimary load case for buckling eigenvalues is scaled with

thebuckling factor.

the other hand, however, the load deformation curve of ashell has in general a reducing curve after the ramificationpoint, that one can imagine from the point A to the point B.The reducing curve cannot be processed currently with themoduleASE.

The critical buckling load for a spherical shell underradial pressure according to the theory of elasticity, i.e.,classical equation of buckling (1) amounts to = 121,16

kN/m . Thevaluefor thesnowload uniformlydistributedon

the horizontal projection is obtained from ' = /cos

167,10kN/ming to

empirical relation (3) amounts to = 24,85 kN/m . The

value for the snow load uniformly distributed on the

horizontalprojectionis obtainedfrom ' = /cos 4,27kN/m

The program calculates a value for the critical buckling

load of 60,38 kN/m (48,301,25) according to the third-order theory without geometrical imperfection.An analysisaccording to the third-order theory with geometricalimperfection gives a critical buckling load value of 23,88

kN/m (19,101,25).

The value of critical load 167,10 kN/m defined byclassical equation, which is based upon the small-displacement theory does not agree with ultimate load

results of 60,38kN/m and23,88kN/m .Thisdiscrepancy isexplained by applying the large-deflection theory of

buckling, which takes into account the squares of thederivatives of the deflection, initial geometricalimperfections and a host of additional factors. The ultimate

load result of 23,88 kN/m is in satisfactory agreement with

theempirical solutionof34,27kN/m .Aplot of a limit load iteration for the node number with

the largest displacement can be generated. Load-deformation curves with and without geometricalimperfections are drawn for the node with the maximumvertical displacement, as shown in Fig. 8. With the firstultimate load iteration (curve A) a ramification problemwithout geometrical imperfection is processed. Thedeformations increase almost linearly; from a specific point(ramification point) no further load increase is morepossible. Curve B shows the load deformation curve withthe geometrical imperfection from the first scaled bucklingmode shape. The ultimate load is smaller now, what on onehand results from the scheduled deformation (u-0) and on

q

q q

q

q q

cr

cr cr

cr

cr cr

2

2

2

2

2

2

2

2

2

2 2

2

2

=

with = 31,62. To relatetheoreticalvalues toactual test data the critical buckling load accord

= 3, where = 31,62.

Buckling analysis of concrete spherical shells I. Mekjavi

Figure 7First buckling shape on deformed structure with geometricalimperfection for Ehime dome

Figure 8Load-deformation curves with and without geometrical

imperfections for the node with the largest vertical displacementin Ehime dome

The effects of the shell rise on the value of the criticalbuckling load are analyzed for the Ehime shell with andwithout geometrical imperfection. The values for diameter

Figure 9Principal membrane forces in the Ehime dome(a) without and (b) with geometrical imperfection


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637Tehni ki vjesnik 18, 4(2011) 633-639,

(span) and thickness of Ehime shell are kept constant. Thedimensions of the ring and columns are equal to 4060 cmand 5050 cm, respectively. The slope of the shell is setequal to 2. The only variable that is changed is rise . Therise variesfrom 7,8 and 9 metersin thisstudy.

Tab. 1 gives the effects of rise on the principalmembrane forces and the buckling load fora spherical shell

with a uniform thickness of 8 cm. It is seen that if theprincipal radii of curvature of the shell surface are larger(that is, the shell is flatter), the membrane forces aregenerally greater. Hence the value of the buckling load willbe lower,possibly substantially lower.Increasingthe rise bycca 30 % increases the buckling load factor by a factor ofabout2.

Typically, applying initial geometrical imperfectionrelated to the first buckling mode, the buckling load for thesystemwithout imperfection is reduced by a factor of about2,5.

In addition, it is seen that the compressive hoop forcesof the Ehime domeunder snowloadchange to tension when

geometrical imperfection in the form of the first bucklingmode shape with a maximum deviation of 20 mm occurs.Fig. 9 shows the principal membrane forces for the Ehimedomewithandwithout geometrical imperfection. Since thisinitial geometrical imperfection leads to tension in hoopdirection it is mandatory to include this possibility in thelayoutof thesteelreinforcementof theshell.

It should be noted that such observation is not verifiedontheEhime domewithrise = 9 m.

Notice also that the buckling of the Ehime shell resultsin excessive principal tensile membrane forces which arerestricted to a narrow zone at the edge of the dome. Thesetensile forces produce too high stresses in the reinforcedconcrete and should be reduced by increasing the edge ring

size (stiffness).

dd

d

d

4.2Interaction of thenon-linearities

The analyzed concrete spherical shells are also used todemonstrate the interaction of the two main non-linearities,materialandgeometrical.

An analysis with non-linear (elasto-plastic) material

was performed simultaneously with a geometrically non-linear analysis.

The material behavior of reinforced concrete can bedescribed by the following properties: non-linear stress-straincurve in tensile andcompressivezone,contributionofthe concrete between cracks (tension stiffening), non-linearmaterial behavior of the steel reinforcement and simplifiedcheckof theshell's shearstress.

The buckling load obtained by a static geometricallynon-linear analysis and a combination of geometrically andmaterially non-linear analysis of spherical shells is given inTab.2.

Forthe Ehimeshell thestableultimateload calculations

geometrically and materially non-linear end now at about11,30 kN/m (9,041,25) and4,48 kN/m (3,581,25) for thesystem without and with geometrical imperfectionsrespectively. Hence,the buckling load in themateriallynon-linear, large deformation analysis is reduced by a factor of5,3.Applyinginitial geometrical imperfection in the formofthe first buckling mode shape with a maximum deviation inglobal X direction of 20 mm, the buckling load for thesystem without imperfection is reduced by a factor of about2,5. An iso-area presentation of the plastified zones leads toFig. 10.

The maximum bottom non-linear von Mises stressesare equal to 12,70 and 7,26 MPa for the system without andwithgeometrical imperfections, respectively. Fig.11showsthe resulting von Mises stress state with high stressconcentrations in the vicinity of the concentrated supports.A slight geometrical deviation (related to the first buckling

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I. Mekjavi Analiza izboavanja betonskih sfernih ljusaka

Table 1Effect of rise on principal membrane forces and buckling load for Ehime domed

Rise d/m Loading System without imperfection System with imperfection

Bucklingload factor

Principal membrane forces/ kN/m

Bucklingload factor

Principal membrane forces/ kN/m

Min.

compressive Max. tensile

Min.

compressive Max. tensile

7 Dead load 54,4 170,0 126,8 162,2

Snow load 48,30 2500,0 5014,0 19,10 2353,0 1872,0

8 Dead load 53,6 147,3 100,1 145,0

Snow load 64,00 2820,0 5576,0 25,50 2330,0 2221,0

9 Dead load 53,8 130,6 59,6 127,1

Snow load 80,30 2944,0 5983,0 37,10 1917,0 2776,0

Table2Buckling load obtained by a geometrically non-linear and a combination of geometrically and materially non-linear analysis of spherical shells

System without imperfection System with imperfection

Buckling load factorNon-linear von

Mises tensile

stress / MPa

Buckling load factorNon-linear von

Mises tensile

stress / MPa

Name ofconstruction

Geometricallynon-linear

Geometrically and materially non-linear

Geometricallynon-linear

Geometrically and materially non-linear

Kresge A. - design 1 2,70 0,78 5,83 3,90 0,78 5,83

Kresge A. - design 2 8,00 0,72 5,06 8,20 0,72 5,06Kresge A. - design 3 14,80 1,29 4,79 15,20 1,29 4,79

Het Evoluon 67,70 1,05 3,04 46,10 1,05 3,04

Ehime Public Hall 48,30 9,04 12,70 19,10 3,58 7,26


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638 Technical Gazette 18, 4(2011) 633-639,

mode) may lead to greater stresses towards the apex of thedome as shown in Fig. 11 (b) compared to (besides the edgebending effect)a homogeneous stress state shown in Fig. 11(a).

In [5] it had been shown that structural optimization ofKresge Auditorium results in a distribution of largerthicknessaroundthesupportsequalto 30cm.

Here the critical buckling load and the von Mises

stresses on the shell surface for three different designs arecompared(Tab.2).

In the original design 1 the concrete shell with auniform thickness of 8,9 cm is strengthened with astiffening beam (2045 cm) around the perimeter of thebuilding, and the concrete class is C30/37 [8]. In the design2 the concrete strength of thedistributed thicknessshell and2045 cm edge beam is C40/50. The design 3 comprisesdistributed thickness shell and (3030 cm to 3070 cm)edgebeamwith higher concretestrengthC45/55.

The maximum bottom non-linear von Mises stressesare equal to 5,83 MPa, 5,06 MPa and 4,79 MPa for thedesign1, design2 anddesign3, respectively(Tab.2).These

maximum von Mises stresses occur in the region of thesupports, gradually decreasing to approximately zero at thecenter of the shell (Fig. 12). It has also been found that thedesign 3 develops less tensile area and smaller maximumtensile stresses obtained by a linear analysis, and thus is amore efficient design. Also, the deflections for thedistributed thickness shells (design 2 and 3) were muchsmaller than the uniform thickness shell (design 1). Inaddition, it can be shown that the edge beam of uniformlyvarying crosssection (height variesfrom30cmatapex to70cm at supports) in design 3 enhances the shell stiffness,reducing maximum (principal) tensile stresses anddeflection, and thereby reducing reinforcements. Also, thehigher concrete strength of C45/55 reduces the deflectionandtheamount of reinforcements.

Here the critical buckling load obtained by ageometrically non-linear analysis is maximized for theKresge shell by structural optimization. Optimization isbased on the structure without geometrical imperfectionleading to a buckling load factor of 14,80 (Tab. 2). Also themaximum geometrical deviation in globalX directionof 20mmwasintroduced inanadditional simulation assuming animperfection shape taken from the first buckling mode. Inthis example, the initial geometrical imperfection(corresponding to the first buckling shape) increases thefinal load level to15,20because it enhances thecurvatureofthe shell roof. Including the two main non-linearities,

material and geometrical, the buckling load is reduced to1,29 showing thestronginfluenceof thenon-linear material

Buckling analysis of concrete spherical shells I. Mekjavi

Figure 10Plastified zones in the Ehime dome with geometricalimperfection resulting from geometrical and material nonlinear

analysis

Figure 11Bottom non-linear von Mises stresses in the Ehime dome(a) without and (b) with geometrical imperfection

Figure 12Bottom non-linear von Mises stresses in the Kresge shell


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behavioron the structural response.This important interaction is also demonstrated by the

Evoluon shell.As canbe foundin[5]structuraloptimizationofEvoluon results ina shell withuniform thickness of8 cm,reinforced with meridional andhoop ribs.Theribbedmodel

built in Sofistik has a 2030 cm ring at the top of the domearound 6,70 m diameter skylight. Radiating off of the ringbeam at the top of the dome are 3060 cm ribs at 7.Addedare two hoop3060cm ribsthat are located at6 m,and12,2m from the edge ring. The edge ring at the junction of theupper and lower shell is 4060 cm with a 77 m diameter. A6080 cm bottom ring is supported by 8080 cm V-shapedcolumns. The lower shell also has two hoop 3060 cm ribsthatarelocatedat6m,and15,4mfromtheedgering.

In this example, the buckling load factor of 67,70obtained by a geometrically non-linear analysis for thestructure without geometrical imperfection is reduced to46,10 by the initial geometrical imperfection related to thefirst bucklingshape (with themaximum deviationof20 mminglobalX direction)showinga substantial sensitivity(Tab.2). Tab. 2 also shows that the buckling load obtained by acombination of geometrically and materially non-linearanalysis is significantly reduced to 1,05. The maximumbottom non-linear von Mises stress is equal to 3,04 MPa.Fig. 13 shows the von Mises stresses on the shell surface forEvoluon.

From Tab. 2 the following conclusions can be drawn:ncluding only one kind of non-linearity does not give a

realistic situation and their combination results in thedecreasing ultimate failure load.

From a detailed structural buckling assessment ofanalyzed concrete spherical shells it can be concluded thatshells can be extremely sensitive with respect to slightdeviation of their ideal parameters like initial geometry,boundary conditions etc. Initial geometrical imperfectionsand non-linearities tend to prevent the most real structuresfrom achieving their unrealistically high failure loads. Toget a more accurate answer nonlinear analysis should becarried out, taking into account geometrical imperfections,material non-linearities, edge effects which cause bendingetc., provided the shell is less prone to a sudden bucklingfailure.

In the finite element buckling analysis of concretespherical shells, consideration shall be given to the possiblesubstantial reduction in the value of the buckling loadcausedby large deflections,material non-linear effects, and

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Conclusion

639Tehni ki vjesnik 18, 4(2011) 633-639,

the deviation between the actual and theoretical shellsurface.

6References

[1] Timoshenko, S. P.; Gere, J. M. Theory of Elastic Stability.SecondEdition.International StudentEdition.McGraw-Hill,

Singapore,1963.[2] Billington, D. P. Thin Shell Concrete Structures. Second

Edition.McGraw-Hill,NewYork- Toronto,1982.[3] Bradshaw, R. Application of the General Theory of Shells.

Journalof theAmericanConcrete Institute. 58,2(1961), 12 9-148.

[4] SofistikSoftwareVersion2010[5

[6] Ford, E. R. The Details of Modern Architecture. Volume1928 to 1988. The MIT Press, Cambridge, Massachusetts,2003.

[7] Rile, H. Prostorne krovne konstrukcije.

[8] http:/www.arche.psu.edu/thinshells/module III/case_study_3.htm

//

] Mekjavi, I.;Piulin,S. StructuralAnalysis and Optimizationof ConcreteSpherical andGroinedShells. // Tehnikivjesnik.4, 17(2010), 537-544.

2:

Graevinska knjiga,Beograd,1977.

Author's addresses:

Doc. dr.sc. IvanaMekjavi,dipl. ing.gra.Universityof ZagrebFacultyofCivil Engineering

Zagreb,Croatiae-mail: [email protected]

Kaieva2610000

I. Mekjavi Analiza izboavanja betonskih sfernih ljusaka

Figure 13Bottom non-linear von Mises stressesin the ribbed Evoluon shell

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Buckling Analysis of Concrete Spherical Shells